Numerical Algorithm for Constructing 3D Initial Stress Field Matching Field Measurements

7/29/2019 Numerical Algorithm for Constructing 3D Initial Stress Field Matching Field Measurements

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1. INTRODUCTION

There are many subsurface applications that require the

knowledge of in-situ stresses. In the petroleum industry,

these are well design, evaluation of the cap rock

integrity, compaction and subsidence calculations. The

virgin in-situ stress can be used directly in a green fielddevelopment and also as an initial state for calculating

the mechanical response to fluids production from or

injections into underground formations.

To estimate the initial stress field it is often assumed that

the stress is uniform in lateral directions and one of the

principal stresses is vertical. Then the vertical and

horizontal stress profiles are estimated by averaging the

available stress indications such as the data obtained

from density logs, leak-off tests, borehole breakouts and

regional stress maps.

There are three major drawbacks of such an approachthat need to be addressed. First, the average stress

profile may not be sufficiently accurate in cases oflaterally heterogeneous fields (e.g., due to complex

surface topology or presence of salt domes). Second, it

does not make full use of the very limited data. And

third, it may not be in equilibrium to be used as the

initial stress distribution in 3D geomechanical modeling.The modeling tools may adjust the uniform stress profile

until it satisfies equilibrium equations. However, the

resulting adjusted initial stress field does not necessarily

satisfy the original field data such as leak-off pressures.

Quantitative estimation of the stress distribution is

extremely challenging due to the lack of data. Not only

are direct stress measurements scarce, but also there are

large uncertainties in the geology (configuration andhistory of deformation) and material properties.

The approach undertaken in this paper is based on the

fact that regardless of the deformation history of a

subsurface, the stress field still must satisfy the

equilibrium equations. With certain assumptions, it may

even be argued that the bulk of the current stress is due

to present external loading, i.e., distributed load (e.g.,

gravity and pressure source) and boundary conditions. If

this is the case then the current in-situ stress can be

estimated by virtually unloading the system.

This follows the original work of McKinnon (2001), [1].

In his paper, McKinnon suggested modeling stress in thesubsurface by applying traction boundary conditions.

These boundary conditions become the main unknown

as they are adjusted to fit the field stress measurements.

However, McKinnon considered mainly mining

applications, in which there were more measurements

than unknowns. In petroleum applications, the situationis exactly the opposite: there are very few data and the

problem is thus heavily under-constrained. In the present

study, this difficulty is addressed by applying an

ARMA 10-324

Numerical algorithm for constructing 3D initial stress field

matching field measurements

Madyarov, A. I. and Savitski, A. A.

Shell Exploration and Production Company, Houston, Texas, USA

Copyright 2010 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 44th

US Rock Mechanics Symposium and 5th

U.S.-Canada Rock Mechanics Symposium, held inSalt Lake City, UT June 2730, 2010.

This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review ofthe paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, ormembers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMAis prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. Theabstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT:A numerical approach for calculating the 3D virgin stress distribution in a subsurface model is developed. A

superposition principle and an inversion technique are used to calculate a consistent stress field that satisfies the available

measurements as well as equilibrium and compatibility equations. An elastic unloading response of the rock is assumed and

applicability of the method for non-linear rocks is discussed. The solution to the forward problem is obtained using the finiteelement method. The ill-posed inverse problem is solved by minimizing a least-squares functional and by using regularization. The

convergence of the algorithm is demonstrated with a numerical example. The work has application in the field of petroleum

geomechanics where large-scale subsurface modeling requires stress initialization. The main advantage of the presented algorithm

is that a consistent three-dimensional initial stress field that matches the available scarce stress data is constructed and can be used

in the subsequent modeling.


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inversion technique with regularization, which is

detailed below.

Although the level of uncertainty involved in such

calculations is fully recognized, the main objective of the

work is to develop a method for determining in-situstress that optimizes the use of the field data and does

not violate the governing equations.

2. PROBLEM FORMULATION

The stress in the earth is a reaction to external loading

and unloading. There are different types of loading and

the reaction can be instantaneous or time and history

dependent. The gravity in a layer-cake depositional

model, indeed, yields a laterally uniform stress

distribution with the maximum principal stress being

vertical. However, there are a number of processes that

lead to stress perturbations: burial, uplift, tectonics,

pressure inflation and depletion, heating and cooling,

and stress relaxation. Regardless of the origin and

history of the stress development, as long as the residualstress (after unloading) is negligible, the current stress in

a field must balance the external loading. These loading

conditions could be distributed (e.g., gravity force, pore

pressure inflation and depletion, or thermal) or applied at

the boundary (tectonics).

The pore pressure and temperature loadings could be

crucial for certain applications. However, including

these effects is straightforward and they are not

considered in this study for simplicity.

Burial, uplift and stress relaxation are all affected by the

rock constitutive behavior. Traditional constitutivemodels consider elastic, plastic and creep responses of

the rock subject to external loading. Models of the rock

deformation over time are non-linear, history- and time-dependent. However, the approach opted in this work is

different. The current stress distribution is estimated not

by following the loading history, but by unloading the

subsurface. The following assumptions are necessary to

justify such an approach:

At the relevant length scale the residual stresses

remaining after complete unloading are negligible

compared to the accuracy of the stress

calculations. All rock formations are assumed to unload linearly

and elastically.

The first assumption is quite strong and may not be

applicable in certain situations. For example, theunloading of two pre-stressed lithological units with full

bonding at their interface and with different properties

will result in residual shear stress near the interface.

However, such residual stress would be localized around

the interface and the solution would still be applicable

away from the interface. In this work, such situations are

not considered. However, modifications to the method of

solution can be made to account for some of the residual

stresses. To relax the second assumption, a constraint

can be imposed that the rock fails at a certain stress state.

However, this is also not considered in the work

presented. Under these two assumptions, the current

stress can be calculated by loading the domain elastically

while using elastic parameters for unloading, since linear

elastic deformation is reversible.

The problem can now be formulated as follows.

Consider a subsurface domain (typically a box withdifferent lithological units as shown in Fig. 1 in 2D). The

stress field to be constructed is such that it satisfies

equilibrium equations and matches available field stress

measurements. There are several sources of the stress

data that can provide estimations of the magnitude and

orientation of the stresses. While it is important to

incorporate all available information for the applications,

this study is limited to using only the leak-off test data

(i.e., estimates of the minimum principal stress).

The next section describes the method of solution wherethe stress field matching the observation data isconstructed as an elastic response to the adjustable

boundary conditions.

gLOT

LOT

LOTLOT

B.c. assumed

Known b.c.

? ?

Fig. 1. Problem formulation. 2D cross-section of the

subsurface model.

3. METHOD OF SOLUTION

3.1. General ApproachFollowing the approach developed in [1]-[3], the total

stress field, , in the earth's crust is decomposed into

gravitational and tectonic parts. This is a formalsuperposition since the tectonic part does not only

account for the tectonic loading per se, but also for the

stress relaxation and any other stress not accounted for

by the gravity term. So, the term tectonic refers to the

mode of application of this load, which is through thelateral boundary conditions.

The gravitational stress tensor field, grav

, is induced bythe gravity load acting on the rock mass (under the fixed

lateral boundary conditions) and can be calculated from

an elastic numerical model on the basis of the


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supposedly known structure of the subsurface and rock

mass density distribution.

The tectonic part, tect, representing a perturbation of the

elastic gravitational stress field is unknown and may

include stresses due to local or global tectonic activity in

the area as well as some other factors. However, it can

be uniquely obtained by specifying the boundary

conditions, since the solution of an elastic problem is

unique.The problem of constructing the in-situ stress is reduced

to an elastic boundary-value problem with unknown

traction boundary conditions and a known solution at

some points inside the domain. The task is to reconstruct

the boundary conditions by inversion of the available

data.

As a result, the total in-situ stress field, , in the modelblock can be found as

(r) = grav

(r) + tect

(r). (1)

The objective of the analysis is to determine the tectonic

traction that must be applied at the side boundaries of the

block to reproduce the stress deduced from the leak-off

test measurements, i.e., the minimum principal stress

1obs,j atMobservation pointsrj:

1[(rj)] = 1obs,j

, j = 1,M. (2)

Here, 1[(rj)] is the minimum eigenvalue of the total

stress tensor at pointrj.

3.2. Gravitational and Tectonic Boundary-Value

ProblemsIn Eq. (1), both gravitational and tectonic parts of the

stress field grav

and tect

are solutions to thecorresponding linear-elastic boundary-value problems in

the modeled subsurface block. In the gravitational

problem, the gravity provides the body-force distribution

and normal tractions may be prescribed at the top surface

of the block to simulate water column pressure loading

on the seabed for subsea applications. In the tectonic

problem, the body force is zero and the top surface is

traction free (Fig. 2).

In both problems, zero normal displacement and zero

tangential traction are prescribed at the bottom

(horizontal) boundary of the block, i.e. free sliding

conditions.

In the gravitational problem, free sliding conditions are

prescribed on the vertical sides of the model (i.e., zeronormal displacement and zero tangential traction).

In the problem for the tectonic stress, traction ttect

(r) is

applied at the sides of the block (see Fig. 2). This

traction is originally unknown and is solved for bymatching the stress from the obtained solution and the

field data. To prevent possible rigid body motion in the

horizontal plane in the tectonic problem, one may fix,

for example, the displacement at the center of the bottom

of the model block and set the normal horizontal

displacement to zero in the middle of one of the bottom

edges.

In addition, the perfect bonding interfacial conditions

(i.e., continuous displacements and tractions) areprescribed at lithological interfaces in both problems.

In the present analysis it is assumed that the principaldirections of the far-field tectonic stress are known, one

of them is vertical, and the horizontal principal direction

does not change with depth. The model block is then

oriented in line with the horizontal principal directions.

According to these assumptions, the unknown tectonic

tractionttect(r) applied at the side boundaries are sought

to be normal to the boundary, varying only with depth

and being the same on the opposite sides of the modeled

block. Therefore, the unknown traction on the side

boundaries can be characterized by profiles of txtect(z)

and tytect(z) with vertical coordinatez.

It is important to note that different assumptions can be

used regarding the distribution of the far-field stresses in

this method. The only requirement is that the total

applied forces are in balance.

3.3. Inverse ProblemTo parameterize the problem of calibrating the tectonic

boundary conditions for numerical models using leak-off

test data, one can consider txtect(z) and ty

tect(z) in the form

of continuous piecewise-linear functions of z. Upon

(a)

x

y

z

O

g

(b)

x

y

z

O

txtect

tytect(a)

x

y

z

O

g

(b)

x

y

z

O

txtect

tytect

Fig. 2. a) Gravitational and b) tectonic boundary-value problems.


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choosing n nodal points zk, the far-field tectonic stress

tensor components can be represented as:

txtect(z;q) = s

=

n

k

kkzQq

1

)( , tytect(z;q) = s

=

+

n

k

kknzQq

1

)( , (3)

where Qk(z) are continuous piecewise-linear shape

functions such that Qk(zj) = kj (Kronecker delta) as

shown in Fig. 3, q = (q1, q2,, q2n) is a vector ofunknown weight coefficients that can be used as the

optimization parameters, and s is a scaling factor. FromEq. (3),

txtect

(zk) = sqk, tytect

(z) = sqn+k, k= 1, n. (4)

z

z1

Q1

z2

z3

zn-1

zn

0

0 1

Q2 Qn txtect

0 1 0 1 q1q2qn

s

z

z1

Q1

z2

z3

zn-1

zn

0

0 1

Q2 Qn txtect

0 1 0 1 q1q2qn

s

Fig. 3. Piecewise-linear shape functions for tectonic tractions.

Due to the linearity of the problem, the tectonic stress

field tect

in the modeled block with side boundary

tractions given by Eq. (3) can be calculated for a

particular choice of vector q as

tect

(r;q) = =

n

k

k

kq

2

1

)(r (5)

where k is the stress distribution due to far-fieldtractions defined by Eq. (3) with vector q such that the

k-th coefficient qk equals one and all other coefficients

are zeros.

In mining applications (see [1]), the measurements of all

components of the total stress tensor might be available,

resulting in a linear algebraic system for the unknown

coefficients. In petroleum applications, it is not the caseand only the minimum principal stress can usually be

deduced from leak-off tests. Although the total stress

tensor depends linearly on parameters q, the problem

is nonlinear due to the known analytical dependence ofthe minimum eigenvalue 1() on the components of.

The problem of calculating the parameter vector q to

match the measurements accordingly reduces to solving

the system (2) in a least-squares sense with respect to q,i.e., to the following minimization problem

F(q) [ ]nR

M

j

j

j 2min);(

1

2,obs

11

=

q

qr , (6)

where is the total stress field defined by Eq. (1) with

the tectonic component tect

depending on parameters q

through Eq. (5). Note that there is no need to solve the

tectonic boundary-value problem directly for different

sets of parameters q during the minimization process.

Rather, quantities k(rj) can be pre-computed in advance

and then linearly combined with coefficients qkaccording to Eq. (5) to obtain the tectonic field at each

iteration. This saves a significant numerical effort

otherwise required for solving the forward problem with

a finite element method.

A gradient-based minimization technique was employed

to find the minimum of the objective functional F. The

particular choice of the minimization tool was left to

function lsqnonlin from MATLAB OptimizationToolbox (the Levenberg-Marquardt method with line

search) [4]. The gradient of F required for the method

can be evaluated analytically, however, the finite-

difference approximation of the derivatives provided by

lsqnonlin gives reasonable results as well.

In a typical setting, there are very few data points

available. Consequently, the number M of equations in

Eq. (2) is usually smaller than the number of unknownsfor parameterization of the tectonic boundary conditions

(in this case, 2n). This fact makes system (2) under-

determined. Such systems may have more than one

solution. In fact, they usually have an entire space of

solutions. Thus, in theory, the minimization problem (6)

may have a manifold of exact solutions bringing the cost

functional F to zero. Therefore, a particular solution

chosen by the minimization algorithm may change

abruptly depending on the measurement uncertainties

and computational errors in the forward model

predictions. To deal with such instability, a

regularization technique is employed.

The idea of regularization is to add a small penalty term

to the objective functional F based on some additional

physical information. For example, Tikhonov

regularization suggests to restrict the norm of the

solution and therefore the additional term is taken in the

form |q|2, where is a small positive regularizationparameter (see, e.g., [5]). In this study, another choice

of the regularization term is adopted such that the

objective functional is

F(q) [ ]=

M

j

j

j1

2,obs

11

);( qr +

+ 2

+++

+=

+

=

+

2

2

12

1

2

1

21

1

2

1)()(

n

n

nk

kkn

n

k

kkqqqqqq . (7)

In Eq. (7), the penalty term represents a sum of

variations of piecewise-linear functions txtect(z; q) and

tytect(z; q), i.e., imposes smooth slowly changing

functions rather than jumping wiggles (see, e.g., [6]). In

this case, the norm of the solution is also restricted as in

Tikhonov regularization, but not so explicitly. It is

important to note that in cases where jumps in far-field


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stress are typical (e.g., in sand-shale sequences), a

different form of the penalty term can be considered. In

fact, it can be constructed such that the profiles with the

correct sequence of high and low stresses are favored

during the minimization process. For the numerical

examples considered, the algorithm with functional

defined in Eq. (7) gave better results than the functional

with Tikhonovs regularization.

The choice of the regularization parameter in Eq. (7)should be such that, on one hand, it is small enough not

to disturb significantly the original functional Fbut, on

the other hand, sufficient to provide a unique global

minimum of F. In this study, is determined aposteriori from the discrepancy principle of Morozov

[6]. If one assumes that an estimate of the error, , in the

measurements (i.e., the right-hand side of Eq. (2)) isknown beforehand, then it is unnecessary to seek an

approximate solution that gives a residual (discrepancy)

in system (2) smaller than . The Morozovsdiscrepancy principle suggests selecting the

regularization parameter = * such that the norm ofthe corresponding discrepancy equals exactly.

4. TEST MODEL

To test the method, without loss of generality, a two-

dimensional plane-strain synthetic model was

considered. The forward models were set up in a finite

element package. In this simple case, the forward

gravitational and tectonic problems can be solved very

fast (in few seconds), since the number of finite elements

needed for the 2D geometry discretization is rather

small. In the plane-strain model, txtect

was taken as the

only non-zero component of the far-field tectonic stress

tensor. Fig. 4 shows the geometry, material parameters

and boundary conditions for the model. The dimensions

of the modeled block were 10 5 km with the origin

located in the middle of the top surface.

To generate synthetic leak-off test measurements, a

piecewise-linear distribution of the far-field tectonic

traction

txtect(z;q) = s

=

n

k

kkzQq

1

true )( , (8)

was taken as the true one. In the numerical example, five

points zk (n = 5) were distributed uniformly with the

depth of the model, and

qtrue

= (q1true

, q2true

,, qntrue

) = (1, 0.8, 0.75, 0.6, 0.2).

Then, the total stress field was generated by summing

the gravitational and tectonic finite element solutions

depicted at Fig. 5. The scaling coefficient s = 10 MPa

was chosen such that the horizontal component xxtect of

the corresponding true tectonic stress was smaller than

the horizontal component xxgrav of the gravitationalstress as in a typical real application. However, the

tectonic stress should still constitute an essential part of

the total stress tensor to make the inversion possible and

worth the effort.

To simulate the noise in the measurements, the total

stress field obtained this way was contaminated with

random noise distributed uniformly within an interval

according to a prescribed multiplicative noise level .Four observation points (M= 4) were taken with

coordinatesx = 1.8 km and z = {2, 4} km, as shown

in Fig. 4 and Fig. 5. The leak-off test measurements at

these points were calculated as the minimum

(a) E1 = 20 GPa1 = 0.35

1 = 2 g/cc

g

E2 = 40 GPa

2 = 0.12 = 3.5 g/cc

(a) E1 = 20 GPa1 = 0.35

1 = 2 g/cc

g

E2 = 40 GPa

2 = 0.12 = 3.5 g/cc

txtect

E1 = 20 GPa

1 = 0.351 = 2 g/cc

E2 = 40 GPa

2 = 0.1

2 = 3.5 g/cc

txtect

(b)tx

tectE1 = 20 GPa

1 = 0.351 = 2 g/cc

E2 = 40 GPa

2 = 0.1

2 = 3.5 g/cc

txtect

(b)

Fig. 4. A two-dimensional test model: a) gravitational and b) tectonic components. (The density of 3.5 g/cc is not realistic and is

used for testing purposes only).

(a) (b)xx

grav, Pa xxtect, Pa

(a) (b)xx

grav, Pa xxtect, Pa

Fig. 5. Synthetic true solution: a) gravitational and b) tectonic components. (Note the difference in color scales.)


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compressive principal stress of the noise-polluted total

field, i.e.,

1obs,j

= 1{[1 + (2 1)][grav

(rj) + tect

(rj)]},

j = 1,M. (9)

where is a random quantity distributed uniformly overthe interval [0, 1].

In the inverse problem, the vector of parameters q was

unknown and to be found using the synthesized leak-off

test measurements (9). Exploiting the fact that the

true value qtrue

of vector q is known for the syntheticmodel, effectiveness of the inversion algorithm can be

directly examined by comparison of the obtained

solution with its true counter-part.

Fig. 6 demonstrates the comparison of the far-field

tectonic stress profile obtained by the data inversion

algorithm described above with the true one. To obtain

the initial guess q0 for vector q required for starting the

iterative minimization algorithm, one can assume that

the total stress tensor is diagonal and does not vary in the

lateral direction. Then the far-field tectonic stress atdepths of observation points can be obtained as

txtect(zj) = 1

obs,j xxgrav(rj), j = 1,M, (10)

as indicated by data points in Fig. 6. The initial guess is

then obtained by fitting a cubic polynomial of the

vertical coordinate z to these data points. Note that the

data points do not fall on the true tectonic tractionprofile because the resulted true stress field is not

laterally uniform due to the complex geometry. The

results shown in Fig. 6 were obtained from the total

stress field polluted with 0.1% noise () when generatingmeasurements by Eq. (9).

The resolved far-field tectonic stress profile depicted in

Fig. 6 is sufficient to reconstruct the stresses inside thedomain that is the main target of this exercise. Therefore,

it is interesting to observe the tectonic stresses at three

vertical cross-sections passing through the two sets of

observation points and the middle of the model block

shown in Fig. 7.Note that the calculated tectonic stress profiles in Fig. 7

(in contrast to the assumed form of the far-field tectonic

stress) exhibit discontinuity at the material interface and

have nonzero values at the top surface. These features of

the elastic model may seem to contradict the

assumptions made for the far-field tectonic stress.

However, looking for a discontinuous far-field tectonic

stress would lead to doubling the number of the

unknown parameters while the influence of such jumps

on the stress field in the region of interest (far away from

the boundaries) would be small. More importantly, the

assumed simple form of far-field tectonic stress stillprovides discontinuous stress profiles inside the region

of interest as can be seen from the plots. Similarly, the

uppermost layers of the earths crust are typically very

Fig. 7. Tectonic stress at internal points of the model block.

0 2 4 6 8 10-5

-4

-3

-2

-1

0

Tectonic tractions tx

tect [MPa]

z[km]

Solution

True

Guess

Data

Fig. 6. Resolved far-field tectonic stress compared with the

true solution.


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soft and do not sustain high compression, relieving the

stress through a local plastic failure mechanism.

As can be seen from Fig. 6 and Fig. 7, for the 2D

example, the suggested inversion scheme provides a

substantially more accurate solution relative to thelaterally uniform approximation (coinciding with the far-

field tectonic stress txtect). This uniform stress is regularly

used in practice as the only available approximation of

the true state of stress but serves just as an initial guessfor the inversion algorithm.

5. CONCLUDING REMARKS

A numerical algorithm is developed for estimating the

in-situ stress conditions in the region of interest based on

the available stress indications such as the interpretation

of leak-off test data. The methodology rests on dividing

the total stress field into the gravitational and tectonic

components such that the gravitational part can be

calculated using the known geometry and materialcharacteristics of the medium, whereas the unknown

tectonic component is determined to match the availablestress field indications. This is an inverse problem,

which, in general, is nonlinear and severely ill-posed due

to the lack of observation data. The inverse problem is

reduced to the minimization of a cost functional with aregularizing penalty term.

The application of the algorithm to a test case shows

promise that the method may give reasonable estimates

for the in-situ stress field that substantially improve the

existing practice.

It is important to emphasize that while the algorithm is

based on an elastic response of the rock it is applicableto real non-linear and non-elastic materials.

Mathematically, the algorithm allows constructing a

consistent initial stress distribution for a 3D subsurface

model such that the equilibrium and compatibility

equations are satisfied while honoring the availablestress measurements. The physical interpretation of this

mathematical approach is the relieving the current stress

state by removing the external forces acting on the rock.

The accuracy of the results relies on the assumptions that

the unloading behavior of the rock is linear elastic andthat the resulting residual stresses after complete

unloading are negligible as compared to the soughtstresses.

There are several important considerations that have

been left out of the scope of this study. Some of the

known problems and a few suggestions are outlined

below.

First, the ill-posedness of the problem can be reduced by

including more field data and physical considerations

into the formulation. In the present realization of the

method, only the LOT data in the form of the minimum

principal stress at a few points are used. In this

formulation, the far-field tectonic stresses perpendicular

to the direction of the minimum principal stress may

have little influence on the value of the minimum

principal stress and thus on the objective functional. This

makes estimating the corresponding components of the

far-field tectonic stress virtually impossible. To

overcome the problem, one may introduce certain

information about the second horizontal principal stress,

for example, into the regularization term. As such

information, one may use a rough estimate of the

maximum-to-minimum horizontal stress ratioR = H/h.The orientation of the breakouts can also be included in

the algorithm by adding the corresponding term to the

cost functional. A breakout may occur as a deformation

of the circular cross-section of the borehole and, if

monitored, may serve as an indication of the localhorizontal principal stress direction. An estimate for the

anisotropy in the horizontal stresses may be obtained

from the breakouts and sonic log correlations.

In this work, only two types of loading were considered:

gravity as a distributed load and tectonic forces viatraction boundary conditions. In practice, it is also

necessary to consider fluid pressure and sometimes

temperature that induce the stress perturbations. This

would be especially relevant when dealing with high-

pressure and high-temperature reservoirs.

Including the pore-mechanical effects is also important

for proper assessment of the rock mass stability. Theimplemented algorithm is based on the elastic model and

does not account for possible failure mechanisms. While

the algorithm does not allow for non-linear deformation,

we may constrain the minimization problem by

forbidding non-physical stress states. Strictly speakingthe effective stress should lie within the failure envelope

of the material. Such a restriction can be implemented in

a hard form, i.e., as a constrained minimization, or in a

soft form through unconstrained minimization of the

cost functional with an additional penalty term. Usually,

the second option is preferable since it provides more

flexibility to the minimization algorithm. The pore

pressure distribution will enter such a constraint through

the definition of the effective stress.

One of the most important issues for inverse problems is

the sensitivity of the solution to variations in the input

data. An attempt to investigate the sensitivity to thenoise in the measurements was made in the present

research through using regularization and the

discrepancy principle. However, it is necessary to run

more tests with more realistic amounts of noise in the

measurements ( 10%). Moreover, it is essential toinvestigate the stability of the solution due to

perturbation of the material and geometric parameters of

the model, as they are also known only up to some

degree of certainty.


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The described regularization technique provides great

flexibility in using various physical considerations in the

analysis. Virtually any information that may be used to

constrain the solution can be expressed in an expression

and supplied to the algorithm through the regularization

(penalty) term. The level of enforcement of these

additional requirements can be manipulated by changing

the regularization parameter based on the level of

confidence in the data. Therefore, the method can be

easily tweaked to fit the special needs of each particularapplication or accommodate new data as they become

available.

ACKNOWLEDGEMENTS

Authors would like to thank Shell for permission to

publish these results. Appreciation is also extended to

John W. Dudley for reviewing this paper and to Kees

Hindriks and Peter Fokker for numerous discussions in

the course of this study.

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[3] McKinnon, S.D., and I. Garrido de la Barra. 2003. Stress

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Numerical Algorithm for Constructing 3D Initial Stress Field Matching Field Measurements